CMU bvh animations and rigging
I have been wrestling with similar problems and I may start my own question, but in the spirit of cooperation and helping this is what I have found:


Rigify and Blender rigs do not work with CMU (without fixing the problems you mention)
Daz 3D rigs also import via FBX unreliably. The best way to get Daz characters into blender for animation is to bring them in via .dae (collada) and then exporting via obj or fbx and use mixamo for animations and for auto-rigging.
You can then retarget mixamo rigs using "bvh retargeter" to CMU - though not reliably. (https://diffeomorphic.blogspot.com/p/bvh-main.html)
N.B. you often get errors "rot &amp; scale" when using bvh retargter. Do not use "transform rot and scale" - ONLY transform scale, otherwise you will get messed up results.
If using MB-LAb, it has it's own bvh importer and they work well with ACCAD bvh - but not so well with CMU. Do not use bvh retargeter.
If importing from MakeHuman, make sure to use the CMU skeleton.
Do not use FBX from MakeHuman to Blender, use the .mhx2 MakeHuman format and get the add-on for Blender for importing MakeHuman assets. You will get a perfectly compliant rig for CMU and BVH retargeter works perfectly with it.


Further notes;


The "rest pose" can look messed up, even though the "pose mode" works fine. I don't know why. In fact, avoid rest mode altogether unless someone else can bob and up tell us what it is useful for and when and why to use it.
Often the bvh's come with a T-pose at the beginning of the animation - it's best to use that, say for adding assets to a character.
"Daz Friendly" bvh's from cgspeed repository don't seem to work - in any configuration I can find, including with Daz 3d studio itself.
If you are struggling to parent an object to your character, you have to be in object mode with the armature in pose position then; object>select armature>switch to pose mode>select bone>parent to bone.


I hope this helps someone. I sure could have used this a week ago. It's a pretty convoluted situation.